With the confession that I've never solo'd a full PT myself, just certain major battles to see if I could, and I've watched some videos of other people soloing ... meaning, my opinion is barely worth squat ...
You can do it with any class or specialization. There is at least one video for each. That said, I think rogue Assassin Archer is going to have the easiest time of it without a lot of cheesing. Tempest Archer, Rift Mage, KE or S&S Champion seem relatively easy as well.
I don't think there is anything wrong with using "passive" exploits, basically, what I'd call "cheesing", for a solo NM run. Like kiting the first Pride demon around the pillar at the Haven temple ruins, or using the fact that the templars in the Hinterlands camp don't aggro if you stay outside of melee range, stuff like that. Some cheesing is going to be necessary in a solo run, unless you are very patient and don't mind reloading a lot. Where I'd draw the line is outright "active" exploits, like jumping into a wall at a critical clipping spot so you are under the map and unreachable.
And you should only need to cheese before Haven. After Haven you should be able to play straight-up and still own every battle, with maybe a couple of boss battle exceptions.
IMO, you can't play a true completionist style 100% solo. There are too many collectibles that require a mage or a rogue, despite the fact that some locked doors can be bashed and some mage "bridges" can be got around on horseback and active exploit of bad clipping. If you bring a party member along just to unlock a door or break a barrier, you're not really playing solo, right?
Best you can do is disable the AI of the party member you bring along, so that they are effectively dead weight in combat if you don't control them. They'll still take some hits intended for you, so still not 100% solo, but if being completionist is more important, maybe that will work for you. Just don't call it a solo run.
Hmm, and I guess you are forced to have either Dorian or Cole in Hushed Whispers/Champions of the Just as well. At least in Hushed Whispers you can disable Dorian's AI. I don't think you get the option with Cole if you side with the Templars.
Now that said, if by completionist you just mean doing every quest possible (not companion personal quests where you are required to have the companion along -- like for Solas, not collectibles, not Codex entries, not special challenges, not exploring every square inch of every map, etc.), that should be do-able solo. I can't think of any quest that absolutely requires a specific class.